Put the ramekins into the dripping tray of hot water in the oven. Cook until the egg white is completely cooked (opaque), about 15 minutes.
Eat as soon as possible, because eggs continue to cook even out of the oven.
Remarks
A good cocotte egg should have the white completely cooked (opaque) and the yolk still liquid, and that's not very easy. A common mistake is to produce a raw egg (white not cooked) or hard (egg yolk cooked). Here are some tips for success:
Eggs must be at room temperature; too cold and they won't cook properly
Watch the cooking carefully through a transparent ramekin
If you have an electronic thermometer, stop cooking when inside the egg white reaches 75°C or 167°F